ESTIMATING PATCH OCCUPANCY WHEN PATCHES
ARE INCOMPLETELY SURVEYED
Leonard A. Stefanski, Matt Rubino and George Hess
∗
Abstract
- Institute of Statistics Mimeo Series #2543
In this paper we describe a method for statistical modeling of the probability of patch
occupancy by a species as a function of patch size and other covariates when patches may be
incompletely surveyed. The model is appropriate for study designs is which each patch: is
visited once; is surveyed at a set of access sites from which an observer can detect the species
within a measurable portion of the patch; and is surveyed until the species is detected or until
all access sites have been visited. The model explicitly accounts for incomplete patch
coverage, providing unbiased estimates of patch occupancy relative to the “naive” modeling
approach that ignores incomplete patch coverage. The statistical model is motivated by and
illustrated with data from a study of Barred owl habitat.
KEY WORDS: Barred owl habitat; biodiversity monitoring; detection probability; logistic
regression; metapopulation dynamics; negative binomial distribution; patch occupancy;
Poisson distribution; statistical modeling.
Institute of Statistics Mimeo Series No. 2543
April 2003
∗
Leonard A. Stefanski is Professor, Department of Statistics, North Carolina State University, Raleigh, NC 27695–
8203. Matt Rubino is Biologist / GIS Analyst, North Carolina Gap Analysis Project, North Carolina State University,
Raleigh, NC 27695-7617. George Hess is Associate Professor, Department of Forestry, North Carolina State University,
Raleigh, NC 27695–8002. Email addresses: [email protected], matt [email protected]. george [email protected].
Stefanski, Rubino, Hess
1
2
Introduction
Population estimates have long been based on counts of individual organisms. However, the
probability of detecting (counting) an individual organism, given that it is present, is almost
always less than one (e.g., Otis et al. 1978; Lancia et al. 1994). Uncorrected counts of individuals
therefore almost always underestimate the true size of a population. Researchers have developed a
variety of statistical modeling approaches to estimate population size when the detection
probability for individuals is less than one (e.g., Seber 1982; Nichols et al. 2000; Williams et al.
2002; Thompson, 1992, Ch 16). These models, with varying levels of sophistication, arrive at
population estimates by adjusting the counts (number of detections) to account for the probability
of detecting individual organisms. In the simplest forms, such as capture-recapture methods, the
count is divided by a single detection probability to provide a corrected population estimate (see
Seber 1982; Lancia et al. 1994). More complex models use continuous detectability functions that
estimate detectability as a function of distance from the observer (e.g., Buckland et al. 1980;
Lancia et al. 1994; Buckland et al. 2001; Williams et al. 2002).
Analogously, the probability of detecting a species in a habitat patch, given that at least one
individual of the species is present in the patch, is usually less than one. Thus, the proportion of
patches in which a species is detected underestimates the true proportion of occupied patches.
Researchers have begun to address the issue of estimating patch occupancy when detection
probability is less than one. For example, McKenzie et al. (2002) developed a method to estimate
patch occupancy that uses a sampling scheme in which patches are visited repeatedly during the
season in which the species is most detectable. Barbraud et al. (2003) developed a method to
estimate patch occupancy for bird colonies that uses a sampling scheme in which patches are
visited from multiple times each year over a number of years.
In this paper, we describe a method for estimating patch occupancy by a species when:
• each patch is visited only once;
• each patch is surveyed at a predefined set of access sites from which an observer can detect
Stefanski, Rubino, Hess
3
the species within a known portion of the patch;
• each patch is surveyed until the species is detected or until all access sites have been visited;
• patches may be surveyed incompletely, even if all access sites have been visited.
Under these conditions, observers might not detect the species in a patch for one of three reasons:
1. the species was not present in the patch;
2. the species was present in unsurveyed portions of the patch, and therefore was not detected;
3. the species was present in the portion of the patch surveyed, but was not detected during the
survey (we assume that this does not happen, i.e., that the species will be detected, if present
in the surveyed portion).
Our estimates of patch occupancy use information about the portions of the patch surveyed at
each access site in addition to the detection status information.
We developed this method as part of an ongoing project to create a wildlife conservation plan for
an approximately one million hectare suburbanizing region in the United States (Hess et al. 2000;
Hess and King 2001; Rubino and Hess 2003). The ability to quickly develop and test habitat
models for a number of species is central to our approach. The habitat models we have been
developing identify habitat patches predicted to be suitable for a species. Our objective in model
verification is to uncover major flaws quickly and inexpensively using a single survey and simple
presence/absence approaches. We are interested in identifying errors of commission (predicting a
species is present when it is not) and omission (not predicting a species is present when it is) in our
habitat model. Patch occupancy of habitat and non-habitat patches is a measure of these errors.
Our method will be of use to those interested in metapopulation dynamics and biodiversity
monitoring programs. Patch occupancy is a key measure in both of these endeavors, in which
patches are almost always incompletely surveyed and species detectability is almost always less
than one. The portion of patches occupied is a state variable in many metapopulation models (e.g.,
Stefanski, Rubino, Hess
4
Levins 1969; Hanski and Gilpin 1997). Change in the occupancy status of patches through time
can be used to estimate metapopulation extinction and recolonization rates (e.g., Barbraud et al.
2003, Mackenzie et al. 2003). In biodiversity monitoring programs, researchers often survey a
portion of an area and use that information to make inferences about a larger area (e.g., Yoccoz et
al. 2001). Clearly, getting the best estimate for patch occupancy is an important goal.
Patch occupancy has been used to define minimum patch sizes for wildlife habitat. For example,
Robbins et al. (1989) used logistic models of forest patch occupancy for 26 area-sensitive bird
species in the Middle Atlantic United States. They reported the patch size at which the probability
of species occurrence was at a maximum, as well as the patch size for which there was at least a
50% probability of finding a species. These patch sizes have been used widely to define minimum
habitat requirements for the 26 area-sensitive bird species documented. The method we present
can improve estimates of species presence upon which such recommendations are based.
2
Survey Protocol
In developing our statistical model, we made the following assumptions:
• Surveys for a species are conducted on a sample of discrete habitat patches of a range of sizes.
• Some of the patches surveyed are non-habitat and not expected to contain the species. These
are included to estimate errors of omission in the habitat model.
• Because of limited access to the patches, most patches are surveyed incompletely.
• Each patch is surveyed from a set of access sites from which the species can be detected.
Access sites are locations within or adjacent to the patch.
• It is possible to measure the total area of the patch.
• The areas of the portions of a patch surveyed from the access sites, and also the areas of
overlap surveyed from multiple sites can be measured.
Stefanski, Rubino, Hess
5
• Access sites may be visited in any order, and the order in which they are visited is recorded.
• The patch survey is considered complete once the species is detected, or after all access sites
have been visited.
We carried out such a survey to test a barred owl (Strix varia) habitat model (Rubino and Hess
2003). We visited a sample of patches delineated by the habitat model. To investigate errors of
omission in the habitat model, we also visited a number of non-habitat patches predicted by our
habitat model to be unsuitable for barred owls. Most patches were on private lands limiting
accessibility. Therefore access sites were established along public roads surrounding the patches
(Figure 1). We used a vocalization call-back technique in which surveyors broadcast a recording of
owl calls in order to elicit responses from territorial individuals (Rubino and Hess 2003). Because
barred owl vocalizations are audible for approximately 800 meters (Smith 1978; Bosakowski et al.
1987), access sites were approximately 1.6 km apart. This spacing was designed to minimize
overlapping sampling effort and maximize patch area coverage per unit effort.
A patch was surveyed until a response was heard, or until all access sites had been visited. Once a
barred owl was detected, sampling was discontinued and the patch was designated as positive for
owl presence. If there was no response at any of the access sites, the patch was scored negative for
barred owl presence. Listening coverage for a patch was almost always incomplete, especially for
larger patches, (Figure 2). For an incompletely surveyed patch, a negative score does not imply
that the patch was empty, only that no owls were present in the surveyed portion.
Rubino and Hess (2002) modeled the relationship between patch size and owl detection using
logistic regression based on data from 95 surveyed patches. This “naive” approach does not use
information about detection probability within patches to estimate the true proportion of patches
occupied. The methods presented here use information about the portion of each patch surveyed to
improve estimates of patch occupancy. Coverage data were available for 94 of the 95 patches Hess
and Rubino surveyed. Because our methods require coverage data, we used only the 94 patches
with coverage information to develop our model. To facilitate comparison, we reproduced Rubino
and Hess’ logistic regression model using data on 94 of the patches (Figure 2, solid curve).
Stefanski, Rubino, Hess
6
A global positioning system was used to obtain the coordinates of each access site during the
survey. We downloaded these coordinates into a geographic information system and quantified the
proportion of the survey patch measured from each access site. We accomplished this by overlaying
the coordinates on the habitat map and assuming an 800m detection radius for each access site.
The cumulative proportions of the patch surveyed at each access site, and the final status of the
patch (owl detected or not), served as the foundation for our statistical model of patch occupancy.
3
Methods: Statistical Modeling and Analysis
3.1
Single Patch Outcome Probabilities
The patch observation process is determined by a sequence of access sites and the portions of patch
covered from each site. Observation of a patch continues until species detection or all access sites
have been visited. For a given patch, let C1 < C2 < · · · < Cm denote the cumulative coverage
proportions afforded by the m access sites in the order that the sites are visited. For example, if 2%
of the patch is covered from the first access site, 3% from the second not covered by the first, and
6% from the third not covered from the first or second, then C1 = 0.02, C2 = 0.05, and C3 = 0.11.
The observation of a patch results in one of m + 1 possible outcomes, AP1 , · · · , APm , AP(0)
m .
• AP1 : The species is detected at the first access site and the site provided a coverage of C1 .
This outcome is summarized by (1, C1 ), where 1 indicates detection and C1 indicates the
proportion of patch surveyed.
• AP2 : The species is detected at the second access site which provided additional coverage of
C2 − C1 . The outcome is summarized by the vector (0, 1, C1 , C2 ). The first two elements (0, 1)
indicate no detection at the first site, detection at the second site. The second two elements
(C1 , C2 ) give the corresponding cumulative proportions of the patch surveyed.
• APj : The species is first detected at the jth access site which provided a coverage proportion
of Cj − Cj−1 not covered by the previous j − 1 sites. The outcome is summarized by the
7
Stefanski, Rubino, Hess
vector (0, . . . , 0, 1, C1 , . . . , Cj ). The j − 1 leading zeros indicate the failure to detect the
species at the first j − 1 sites; (C1 , . . . , Cj ) indicate the corresponding cumulative proportions
of the patch covered.
If all access sites are visited, there are two possible outcomes at the final site.
• APm : The species is first detected at the mth access site, summarized
(0, . . . , 0, 1, C1 , . . . , Cm ) where the 1 appears in the mth component.
• AP(0)
m : The species is not detected at any of the m sites, summarized (0, . . . , 0, C 1 , . . . , Cm ).
Under these assumptions, we derived the probability of each outcome. First, we determined
conditional probabilities given that there are k individuals of the species in the patch.
Unconditional probabilities are obtained by summing over an assumed distribution for the number
of individuals per patch. We refer to the successive new portions of patch surveyed at each access
site as subpatches. The mathematical assumptions used to develop our model are stated in this
section. Their reasonableness, and the likely consequences of violations, are discussed in detail in
Section 5
When the species does not occupy the patch (k = 0), the only possible outcome (assuming the
³
´
(0)
probability of a false detection is 0) is AP(0)
= 1.
m . Thus Pr (APj ) = 0, j = 1, . . . , m, and Pr APm
Probabilities when k > 0 are derived under the assumptions that individuals are uniformly and
independently distributed throughout the patch, and when present in a surveyed area, individuals
are detected with probability one. Event AP1 occurs if and only if there is at least one individual
in the first subpatch (with proportion C1 ) surveyed. This is calculated as the complement of the
probability that there are no individuals in the first subpatch, and under our assumptions is
Pr(AP1 ) = 1 − (1 − C1 )k . Here, 1 − C1 is the proportion of the patch not surveyed (i.e., all but the
first subpatch). Raising this the kth power gives the probability that all k individuals in the patch
are in the unsurveyed portion (thus (1 − C1 )k is the probability of no individuals in the first
subpatch); and subtraction from 1 results in the probability that not all individuals are in the
8
Stefanski, Rubino, Hess
unsurveyed portion, which is equivalent to having at least one individual in the surveyed portion
(the first subpatch).
Event AP2 occurs if and only if there are no individuals in the first subpatch and there is at least
one in the second subpatch. The probability Pr (AP2 ) can be factored as the conditional
probability of at least one individual in the second subpatch given no individuals in the first
subpatch, and the probability of no individuals in the first subpatch. The probability of no
individuals in the first subpatch was determined previously and is (1 − C1 )k .
The conditional probability of at least one individual in the second subpatch given no individuals
in the first subpatch is calculated using the same reasoning that was used to calculate Pr(AP 1 ).
The area surveyed from the first access site is “removed” from consideration and the remaining
area is regarded as a “new” patch, with the understanding that proportions must be normalized by
the factor 1 − C1 . The conditional probability of no individuals in the second subpatch given no
individuals in the first subpatch is thus {(1 − C2 )/(1 − C1 )}k , and hence the conditional
probability of at least one individual in the second subpatch given none in the first subpatch is
1 − {(1 − C2 )/(1 − C1 )}k . Multiplying by the probability of no individuals in the first subpatch,
(1 − C1 )k , results in Pr (AP2 ) = (1 − C1 )k − (1 − C2 )k . Similar reasoning shows that
Pr (APj ) = (1 − Cj−1 )k − (1 − Cj )k for j = 1, . . . , m. Finally, the probability of no individuals in
the total surveyed area is calculated as the probability that all k individuals are in the unsurveyed
³
´
area, and thus Pr AP(0)
= (1 − Cm )k . The probabilities for all cases can be written
m
Pr (APj ) = (1 − Cj−1 )k − (1 − Cj )k , j = 1, . . . , m,
³
´
Pr AP(0)
= (1 − Cm )k ,
m
k ≥ 0,
(1)
where by definition C0 = 0.
Assuming that the distribution of the number of individuals, K, in a patch is given by a
parametric mass function p(k; λ) = Pr(K = k; λ) depending on the parameter λ, unconditional
probabilities are obtained as
Pr (APj ) =
∞ n
X
k=0
o
(1 − Cj−1 )k − (1 − Cj )k p(k; λ),
j = 1, . . . , m
9
Stefanski, Rubino, Hess
³
Pr AP(0)
m
´
=
∞
X
(1 − Cm )k p(k; λ).
k=0
These probabilities can be written in terms of the probability generating function of K,
P(t, λ) = E(tK ), 0 ≤ t ≤ 1,
Pr (APj ) = P(1 − Cj−1 , λ) − P(1 − Cj , λ),
³
Pr AP(0)
m
´
j = 1, . . . , m,
= P(1 − Cm , λ).
Note that the probabilities depend only on whether an individual was observed (Y = 1) or not
(Y = 0), and on the last two cumulative proportions, denoted (CL−1 , CL ), leading to the concise
representation for the probability of the outcome (CL−1 , CL , Y ),
Pr (CL−1 , CL , Y ) = {P(1 − CL−1 , λ) − P(1 − CL , λ)}Y {P(1 − CL , λ)}1−Y
(
P(1 − CL−1 , λ)
−1
= P(1 − CL , λ)
P(1 − CL , λ)
)Y
for Y = 0, 1, and 0 ≤ CL−1 < CL ≤ 1.
Different choices for the distribution of K (the number of individuals per patch) result in different
versions of the models. We continue development of the model in parallel for the cases in which K
is Poisson(λ) and Negative Binomial(r, λ). The Poisson distribution is the most commonly used
model for count data and thus it is a natural choice; the Negative Binomial is included for
comparison and to illustrate the generality of the modeling approach. For the Poisson distribution,
P(t, λ) = exp{−λ(1 − t)} and
Pr (CL−1 , CL , Y ) = exp(−λCL ) {exp(λ (CL − CL−1 )) − 1}Y .
For the Negative Binomial distribution, P(t, λ) = {(1 − λ)/(1 − λt)} r and
Pr (CL−1 , CL , Y ) =
(
1−λ
1 − λ(1 − CL )
)r ("
1 − λ(1 − CL )
1 − λ(1 − CL−1 )
#r
−1
)Y
.
10
Stefanski, Rubino, Hess
3.2
Incorporating Predictor Variables
The number of individuals in a patch will depend on patch characteristics and such dependencies
are incorporated through the parameter λ. We consider patch size, although the modeling extends
readily to other predictor variables. For example, if the mean number of individuals is thought to
be proportional to patch size, then when K ∼ Poisson(λ), λ = βX is appropriate, where β is an
unknown regression parameter and X is patch size. More generally, λ = λ(X, β), some function of
patch size X and parameter vector β.
Logistic regression is a common model for relating patch occupancy to habitat characteristics such
as patch size (Rubino and Hess, 2003; Scott et al., 2002). In our modeling framework, patch
occupancy corresponds to K > 0, and thus the probability of patch occupancy is
Pr(K > 0) = 1 − Pr(K = 0) = 1 − P(0, λ(X, β)). It is possible to choose λ(X, β) so that the
induced model for patch occupancy is logistic. This facilitates comparison of logistic models fit
under the assumption of perfect detection (sometimes called “naive” models) to our models that
explicitly account for imperfect detection due to incomplete coverage. For the model with K
distributed Poisson,
λ(X, β) = ln {1 + exp(β0 + β1 X)}
(2)
induces a logistic model for patch occupancy, Pr (K ≥ 1) = F (β0 + β1 X) where F is the logistic
distribution function, F (t) = 1/ {1 + exp(−t)}. For the model with K distributed Negative
Binomial,
λ(X, β) = 1 − [1/ {1 + exp(β0 + β1 X)}]1/r
(3)
induces a logistic model for patch occupancy.
3.3
Likelihood Methods
When a sample of n patches are studied resulting in data {(Xi , CL−1,i , CL,i , Yi ) ,
the log-likelihood for β is L (β) =
Pn
i=1
i = 1, . . . , n},
Li (β) where, with λi = λ(Xi , β),
Li (β) = Yi ln {P(1 − CL−1,i , λi ) − P(1 − CL,i , λi )} − (1 − Yi ) ln {P(1 − CL,i , λi )} .
11
Stefanski, Rubino, Hess
b is found by numerical maximization of L(β). Inference is
The maximum likelihood estimator β
b The
generally based on the large-sample, normal approximation to the distribution of β.
b can be estimated by either the inverse sample information
asymptotic-distribution variance of β
³
matrix VbINFO = n−1 Ab−1 or the so-called sandwich variance matrix VbSAND = n−1 Ab−1 Bb Ab−1
n
1X
b L̇ (β)
b T,
b
B=
L̇i (β)
i
n i=1
³
1
Ab = −
n
X
n i=1
´
´T
, where
b
L̈i (β),
(4)
L̇i (β) = (∂/∂β) Li (β), and L̈i (β) = ∂/∂β T L̇i (β). The variance estimate VbINFO is preferred
when the assumed model is correct, whereas the alternative estimate VSAND is robust to model
departures (Stefanski and Boos, 2002).
3.4
Computation and Software
Fitting the proposed models to data involves standard statistical computations. We have written
programs in GAUSS (Aptech Systems) for doing the necessary calculations. These programs were
used to compute the estimates reported in the next section. The GAUSS code, and the used in the
following section, are available from the first author upon request.
4
Results: Statistical Modeling and Analysis
We illustrate our statistical models using data from the Barred owl study of Rubino and Hess
(2003). They presented a logistic regression model of observed patch occupancy, i.e., their analysis
did not account for incomplete patch coverage. Observed patch occupancy can differ from true
patch occupancy. Our models are designed to provide improved estimates of true patch occupancy
by explicitly accounting for incomplete coverage. We compare the standard logistic model (ignoring
incomplete coverage) to models from Section 3 In our models we use X = ln(patch size + 1) to
facilitate comparison to the results presented by Rubino and Hess (2003).
Examination of the data revealed some large patches with no observed occupancy (Y = 0) and
coverages much less than 100%. It is not unreasonable to expect that had these large patches been
surveyed completely, owls would have been detected in some or all of them. It follows that the
Stefanski, Rubino, Hess
12
effect of these points on the fitted naive logistic regression model is to bias downward the estimated
probability of patch occupancy for large patch sizes, which tend to have a smaller proportion of
their areas surveyed. Consequently, the logistic model of observed patch occupancy is biased low
for large patch sizes. The models in Section 3 explicitly account for incomplete coverage and should
result in estimates of patch occupancy parameters that are not biased due to incomplete coverage.
We estimated the occupancy curve as function of ln(patch size + 1) based on several choices for
the distribution of the number of individuals per patch, K (Table 1). The naive model is standard
logistic regression of observed patch occupancy, Y , on X = ln(patch size + 1) and coincides with
the results reported by Rubino and Hess (2003), with the previously noted exception that we used
data on only ninety-four patches. The Poisson and Negative Binomial (r = 1, 2, 4) are the models
described in Section 3 with λ(X, β) chosen so that the induced models for patch occupancy are
logistic. The adjusted-coverage models have intercepts (βb0 ) that are less than the intercept of the
logistic model, and slopes (βb1 ) that are greater than that of the logistic model. These differences
are consistent with the expectation that the coverage-adjusted models have higher estimated
occupancy probabilities for large patches than does the naive logistic model. This fact is illustrated
in Figure 2, which displays the estimated patch occupancy models for the naive, Poisson and
Negative Binomial (r = 1) models. All of these models have two parameters so that AIC =
−2(Log-likelihood)+2. The Negative Binomial models fit the barred owl data marginally better
than the Poisson model.
For additional comparison we fit a standard logistic model to data that were modified to represent
the data that might have been obtained with more complete patch coverage. There are five large
patches (patch size > 934ha) in the data set with no owl observed (Y = 0) and low coverage
proportions (< 0.56). Under the supposition that these patches were truly occupied by owls, and
that Y = 0 only because of the low coverage values, we set Y = 1 for these five patches and refit
the standard logistic regression model to the modified data. The estimates from this model appear
in Table 1 under the heading of a modified logistic model, and the corresponding occupancy curve
is shown in Figure 2. The similarity of the modified-data estimates to the coverage-adjusted model
Stefanski, Rubino, Hess
13
estimates is noteworthy.
5
Discussion
We have presented a statistical model to estimate patch occupancy when patches are incompletely
surveyed. Our motivation for developing this method was to improve our ability to validate a
habitat model being used for conservation planning. Compared to the naive occupancy model (i.e.,
proportion of patches in which we detected barred owls) the incomplete coverage model indicates
that our habitat model has fewer errors of omission at larger patch sizes.
For metapopulation modeling and environmental monitoring applications, our incomplete coverage
model provides a better estimate of patch occupancy than simply counting the number of patches
in which a species is detected. In metapopulation modeling, the proportion of occupied sites is a
state variable; in fact, it describes completely the state of the system in the simplest
metapopulation models. Naive approaches that do not account for detection probability less than
one underestimate the proportion of occupied sites. Similarly, monitoring programs require
inferences about areas that cannot be completely surveyed. Naive approaches would again result in
underestimates of the occurrence of species.
Where patch occupancy has been used to define minimum patch sizes for wildlife habitat (e.g.,
Robbins et al. 1989), the naive approach misrepresents the minimum patch size. For the barred
owl, and for all but the lowest occupancy probabilities, the naive approach overestimates the patch
size required for a given probability of occupancy (Figure 2). This has important implications for
conservation reserve planning initiatives — patches previously considered too small to serve as
conservation habitat might be adequate. If one were to choose a minimum patch size for barred owl
habitat based on our data, the incomplete coverage models suggest smaller patch sizes than does
the naive model. For example, for the patch size at which there is a 50% chance of finding a barred
owl (Robbins et al. 1989) our incomplete coverage model (Neg Bin (r = 1)) results in a smaller
estimated patch size (≈ 164ha) than does the naive model (≈ 288ha).
Stefanski, Rubino, Hess
14
Our model relies on a number of assumptions, all of which are likely to be violated to some extent
in applications. The assumptions are of three types: (1) assumptions on the dispersion of
individuals in a patch; (2) assumptions on the subpatch detection method; and (3) statistical
modeling assumptions. We review the key assumptions and give a qualitative assessment of the
effects of violations. Future work on the statistical model will study the impact of model violations
quantitatively, and describe extensions of the basic model that address certain model violations.
The key assumptions on the species under study is that individuals are independently and
uniformly distributed throughout the patch. This assumption is not necessarily violated for species
whose members occur in pairs, packs, flocks, or other family or social units. In such cases the
model applies with the understanding that “individual” means a family or social unit. For
example, for the barred owl data a detection is regarded as a detection of a nesting pair.
The uniformity assumption is likely to be reasonable when habitat quality does not vary
excessively throughout a patch. For the barred owl study, patches were identified in such a way
that ensured a fair degree of habitat uniformity throughout a patch.
The appropriateness of the independence assumption depends on whether individuals tend to
aggregate or be self-avoiding. Self-avoiding species will, on average, be more evenly dispersed than
would be expected under the independence assumption. This means that, relative to the case of
independence, not as much area would have to be surveyed (on average) before an individual is
encountered. Individuals will be more likely to be detected at one of the first few access sites
relative to the case when individuals are independently dispersed, and observed population density
will likely be higher than what would be predicted by a model that assumes independence. This
suggests that estimates of occupancy probabilities derived from a model that assumes
independence might be biased high. The opposite (occupancy probabilities biased low) will likely
be true of gregarious species. With regard to the barred owl study, mating pairs compete for a
limited resource (habitat patches) within the larger landscape matrix of the study area (Research
Triangle) and are therefore territorial. As a result, there is a tendency for pairs to avoid one
another. Thus pairs may be more evenly distributed throughout patches and co-occur less
Stefanski, Rubino, Hess
15
frequently than would be the case under independence assumption. The impact of this possible
violation of the assumptions is not known at this time. Future work is planned to investigate the
robustness of our model to violations of the independence assumption.
The assumptions on the subpatch survey detection method are threefold. The first two are that the
probability of a false detection is zero, and the probability of detection is one when an individual is
in a surveyed subpatch. Violations of the latter are more common and hence more problematic
than are violations of the former, and result in underestimation of occupancy probabilities. Our
basic model can be modified to handle both types of errors provided the error rates of the
detection method are known or can be estimated. The modifications are technically involved and
will be presented in a forthcoming paper. The required error rates can be estimated by multiple
surveys of the same access sites, or experimentally by repeatedly surveying subpatches that are
known to contain individuals (to estimate the probability of detection) or known not to contain
individuals (to estimate the probability of false detection). The third assumption is that subpatch
coverage areas can be determined with acceptable accuracy. Subpatch areas are accumulated over
access sites and divided by total patch area to obtain CL−1 and CL , the patch coverage proportions
required in our model. Thus errors in the subpatch areas are propagated to CL−1 and CL . For the
barred owl study, coverage areas were determined by GIS based on GPS-determined locations that
are accurate to within seven-to-ten meters, resulting in very accurate subpatch coverage area
measurements and thus errors in CL−1 and CL are not likely to be problematic. This may not be
the case in other applications and future work will investigate the effect of errors in C L−1 and CL .
The key modeling assumption relates to the distribution for the latent variable, K, the number of
individuals in a patch. There is very little information in a single data set on which to base the
choice. However, the models fit to the barred owl data suggest that the choice is not too critical.
The Negative Binomial distribution is heavier tailed and more dispersed than the Poisson
distribution, yet the parameter estimates and patch occupancy curves displayed in Table 1 and
Figure 2 are similar, indicating a reasonable measure of insensitivity to the model for K.
Stefanski, Rubino, Hess
6
16
Acknowledgements
We thank Jim Nichols for encouragement and advice in the initial stages of our work, and for
comments on the manuscript. The manuscript also benefitted from the thoughtful comments of
Cavell Brownie and Ken Pollock.
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18
Stefanski, Rubino, Hess
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Figure 1: Portion of study area showing Patches Nos. 26746 and 27295 (gray), access sites along
public roads, survey listening areas, and subpatch coverages (lined). Access site 105015 provided
≈ 99% coverage of Patch No. 27295; the other access sites combined provided ≈ 75% of Patch No.
26746.
19
Stefanski, Rubino, Hess
βb0
βb1
−2(Log-likelihood)
Logistic (naive)
-3.40 (0.82)
0.60 (0.15)
97.34
Poisson
-4.07 (0.96)
0.84 (0.18)
207.27
Neg Bin (r = 1)
-3.83 (0.84)
0.75 (0.14)
198.72
Neg Bin (r = 2)
-4.05 (0.92)
0.82 (0.16)
201.61
Neg Bin (r = 4)
-4.09 (0.94)
0.83 (0.17)
204.10
Logistic (modified) -4.13 (0.97)
0.79 (0.17)
87.02
Model
Table 1: Parameter estimates, standard errors (in parentheses) and log-likelihoods for the naive
logistic model; several models of the distribution of the number of individuals per patches (K) from
Section 3; and the logistic model fit to modified data in which the five largest patches in which no
owls were detected were assumed to have owls present (but not detected because of low coverage).
Stefanski, Rubino, Hess
20
Figure 2: Top. Patch coverage fractions by transformed patch size (non-habitat patches excluded).
Bottom. Estimated probability of patch occupancy. Naive, logistic regression with no adjustment
for incomplete coverage; Poisson, coverage-adjusted Poisson model; NB(r = 1), coverage-adjusted
Negative Binomial model with r = 1; Modified Data, logistic regression with modified data.